You know that is a group homomorphism, and that has order Let's use this:(where is the identity element of G)

Moreover, since is a homomorphism,

So we get What can you deduce?

Now, the second question. In a group, elements orders always divide the group order. If then the only common positive divisor between and will be . For any in we want to prove that i.e. that has order . Some idea? (Using the first result of course)

Since m|k, k=mx where x is a positive integer.
By Lagrange's theorem, mx | |G| and m | |H|. Since |G| and |H| are coprime, there is no common factor other than 1. Thus, m is 1. We conclude that for all .